[FOM] Paolo Mancosu's monograph on 17th century mathematics

Martin Davis martin at eipye.com
Mon Mar 24 17:17:20 EST 2003

I would like call the attention of FOMers to Paolo Mancosu's excellent 
CENTURY. I believe that by seeing how mathematicians grappled with problems 
with infinity at this time, we can get valuable perspective on the 
foundational issues of our day.

I found most striking the general reaction to a result by Torricelli (the 
scientist who took over Galileo's chair in Florence). Torricelli computed 
the finite volume enclosed by a particular infinite surface of revolution. 
(In modern terms: the solid is formed by rotating about the X-axis the 
rectangular hyperbola y = 1/x with x >= 1.)  This result (today a homework 
problem in a standard calculus course) created a sensation. I recommend to 
all Mancosu's discussion of the way in which Torricelli's result played 
havoc with the received wisdom of the day about the actual infinite.

I would like to suggest that mathematical practice (and especially the 
formalisms which tend to give it much of its power) is by its nature 
expansive, that concepts and methods tend to "overspill" leading to 
FOUNDATIONAL questions about the status and validity of the new domains 
hesitatingly revealed. The history of mathematics is replete with examples. 
I believe that it is the work on extensions of ZFC, and especially on large 
cardinals that is the main contemporary example of this phenomenon.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
                          (Add 1 and get 0)

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